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### 17.8 Coordinate Transformations

: [theta, r] = cart2pol (x, y) ¶
: [theta, r, z] = cart2pol (x, y, z) ¶
: [theta, r] = cart2pol (C) ¶
: [theta, r, z] = cart2pol (C) ¶

  Transform Cartesian coordinates to polar or cylindrical coordinates.

  The inputs x, y (, and z) must be the same shape, or scalar. If called with a single matrix argument then each row of C represents the Cartesian coordinate pair (x, y) or triplet (x, y, z).

  The outputs theta, r (, and z) match the shape of the inputs. For a matrix input C the outputs will be column vectors with rows corresponding to the rows of the input matrix.

  theta describes the angle relative to the positive x-axis measured in the xy-plane.

  r is the distance to the z-axis (0, 0, z).

  z, if present, is unchanged by the transformation.

  The coordinate transformation is computed using:

      theta = arctan (y / x)
      r = sqrt (x^2 + y^2)
      z = z

  Note: For MATLAB compatibility, this function no longer returns a full coordinate matrix when called with a single return argument.

  See also: pol2cart, cart2sph, sph2cart.

: [x, y] = pol2cart (theta, r) ¶
: [x, y, z] = pol2cart (theta, r, z) ¶
: [x, y] = pol2cart (P) ¶
: [x, y, z] = pol2cart (P) ¶

  Transform polar or cylindrical coordinates to Cartesian coordinates.

  The inputs theta, r, (and z) must be the same shape, or scalar. If called with a single matrix argument then each row of P represents the polar coordinate pair (theta, r) or the cylindrical triplet (theta, r, z).

  The outputs x, y (, and z) match the shape of the inputs. For a matrix input P the outputs will be column vectors with rows corresponding to the rows of the input matrix.

  theta describes the angle relative to the positive x-axis measured in the xy-plane.

  r is the distance to the z-axis (0, 0, z).

  z, if present, is unchanged by the transformation.

  The coordinate transformation is computed using:

      x = r * cos (theta)
      y = r * sin (theta)
      z = z

  Note: For MATLAB compatibility, this function no longer returns a full coordinate matrix when called with a single return argument.

  See also: cart2pol, sph2cart, cart2sph.

: [theta, phi, r] = cart2sph (x, y, z) ¶
: [theta, phi, r] = cart2sph (C) ¶

  Transform Cartesian coordinates to spherical coordinates.

  The inputs x, y, and z must be the same shape, or scalar. If called with a single matrix argument then each row of C must represent a Cartesian coordinate triplet (x, y, z).

  The outputs theta, phi, r match the shape of the inputs. For a matrix input C the outputs will be column vectors with rows corresponding to the rows of the input matrix.

  theta describes the azimuth angle relative to the positive x-axis measured in the xy-plane.

  phi is the elevation angle measured relative to the xy-plane.

  r is the distance to the origin (0, 0, 0).

  The coordinate transformation is computed using:

      theta = arctan (y / x)
      phi = arctan (z / sqrt (x^2 + y^2))
      r = sqrt (x^2 + y^2 + z^2)

  Note: For MATLAB compatibility, this function no longer returns a full coordinate matrix when called with a single return argument.

  See also: sph2cart, cart2pol, pol2cart.

: [x, y, z] = sph2cart (theta, phi, r) ¶
: [x, y, z] = sph2cart (S) ¶

  Transform spherical coordinates to Cartesian coordinates.

  The inputs theta, phi, and r must be the same shape, or scalar. If called with a single matrix argument then each row of S must represent a spherical coordinate triplet (theta, phi, r).

  The outputs x, y, z match the shape of the inputs. For a matrix input S the outputs are column vectors with rows corresponding to the rows of the input matrix.

  theta describes the azimuth angle relative to the positive x-axis measured in the xy-plane.

  phi is the elevation angle measured relative to the xy-plane.

  r is the distance to the origin (0, 0, 0).

  The coordinate transformation is computed using:

      x = r * cos (phi) * cos (theta)
      y = r * cos (phi) * sin (theta)
      z = r * sin (phi)

  Note: For MATLAB compatibility, this function no longer returns a full coordinate matrix when called with a single return argument.

  See also: cart2sph, pol2cart, cart2pol.

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